On the homeomorphism of spaces and Magill-type theorems (Q2710545)

Motivated by work of K. D. Magill, the authors investigate connections between the homeomorphism relation of spaces and the isomorphism relation of some posets of continuous mappings of spaces. In the first section results are developed which show that the fact that some posets of quotient mappings of two \(T_1\)-spaces are isomorphic implies that the two spaces are homeomorphic. An analogous result is also developed for posets of perfect mappings of Hausdorff spaces. In the second section these results are used to obtain generalizations of theorems proven by K. D. Magill and M. C. Rayburn. The two sample results stated below communicate the flavour of this section. In what follows, \(X\) is a Tychonoff space, \(K(X)\) is the poset of all Hausdorff compactifications of \(X\) and \(rK(X)\) is the set of all spaces which are homeomorphic to \(cX-X\) for some \(cX\in K(X)\). For \(eX\in K(X)\), \(K(eX)= \{cX\in K(X):cX\leq eX\}\).NEWLINENEWLINENEWLINETheorem 20. For locally compact Tychonoff spaces \(X,Y\) and their Hausdorff compactifications \(eX\), \(eY\), the remainders \(eX \setminus X\) and \(eY\setminus Y\) are homeomorphic if and only if the posets \(K (eX)\) and \(K(eY)\) are isomorphic.NEWLINENEWLINENEWLINECorollary 23. Let \(X\) and \(Y\) be locally compact Tychonoff spaces such that \(X\) has a perfect mapping onto \(Y\) and \(Y\) has a perfect mapping onto \(X\). Then \(rK(X)=rK(Y)\).